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Encyclopedia > Euclidean space

Around 300 BC, the Greek mathematician Euclid laid down the rules of what has now come to be called "Euclidean geometry", which is the study of the relationships between angles and distances in space. Euclid first developed "plane geometry" which dealt with the geometry of two-dimensional objects on a flat surface. He then went on to develop "solid geometry" which analyzed the geometry of three-dimensional objects. All of the axioms of Euclid have been encoded into an abstract mathematical space known as a two- or three-dimensional Euclidean Space. These mathematical spaces may be extended to apply to any dimensionality, and such a space is called an n-dimensional Euclidean space or an n-space. This article is concerned with such mathematical spaces. Euclid, is also referred to as Euclid of Alexandria, (Greek: , about 330 BC– about 275 BC), a Hellenistic mathematician, who lived in the city of Alexandria, Egypt, almost certainly during the reign of Ptolemy I (323 BC–283 BC), is often considered to be the father of geometry. His most... Euclid Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. ...


In order to develop these higher dimensional Euclidean spaces, the properties of the familiar Euclidean spaces must be very carefully expressed and then extended to an arbitrary dimension. Although the resulting mathematics is rather abstract, it nevertheless captures the essential nature of the Euclidean spaces we are all familiar with.


An essential property of a Euclidean space is its flatness. Other spaces exist that are not Euclidean. For example, the surface of a sphere is not a Euclidean space, nor is the four-dimensional spacetime described by the theory of relativity when gravity is present. Two-dimensional analogy of space-time distortion described in General Relativity. ... Gravity is a force of attraction that acts between bodies that have mass. ...

Contents

Intuitive overview

One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance and angle. For example, there are two fundamental operations on the plane. One is translation, which means a shifting of the plane so that every point is shifted in the same direction and by the same distance. The other is rotation about a fixed point in the plane, in which every point in the plane turns about that fixed point through the same angle. One of the basic tenets of Euclidean geometry is that two figures (that is, subsets) of the plane should be considered equivalent (congruent) if one can be transformed into the other by some sequence of translations and rotations. (See Euclidean group.) In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ... Point can refer to: Look up Point in Wiktionary, the free dictionary // Mathematics In mathematics: Point (geometry), an entity that has a location in space but no extent Fixed point (mathematics), a point that is mapped to itself by a mathematical function Point at infinity Point group Point charge, an... In Euclidean geometry, translation is a transformation of Euclidean space which moves every point by a fixed distance in the same direction. ... A sphere rotating around its axis. ... A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ... See also: congruence relation In geometry, two shapes are called congruent if one can be transformed into the other by a series of translations, rotations and reflections. ... In mathematics, the Euclidean group is the symmetry group associated with Euclidean geometry. ...


In order to make all of this mathematically precise, one must clearly define the notions of distance, angle, translation, and rotation. The standard way to do this, as carried out in the remainder of this article, is to define the Euclidean plane as a two-dimensional real vector space equipped with an inner product. For then: In mathematics, the real numbers may be described informally in several different ways. ... In mathematics, a vector space (or linear space) is a collection of objects (known as vectors) which may be scaled and added; all linear combinations of vectors are themselves vectors. ... In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ...

  • the vectors in the vector space correspond to the points of the Euclidean plane,
  • the addition operation in the vector space corresponds to translation, and
  • the inner product implies notions of angle and distance, which can be used to define rotation.

Once the Euclidean plane has been described in this language, it is actually a simple matter to extend its concept to arbitrary dimensions. For the most part, the vocabulary, formulas, and calculations are not made any more difficult by the presence of more dimensions. (However, rotations are more subtle in high dimensions, and visualizing high-dimensional spaces remains difficult, even for experienced mathematicians.) Look up vector in Wiktionary, the free dictionary. ... 3 + 2 with apples, a popular choice in textbooks Addition is the basic operation of arithmetic. ... To meet Wikipedias quality standards, this article or section may require cleanup. ...


A final wrinkle is that Euclidean space is not technically a vector space but rather an affine space, on which a vector space acts. Intuitively, the distinction just says that there is no canonical choice of where the origin should go in the space, because it can be translated anywhere. In this article, this technicality is largely ignored. In mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space. ... In mathematics, a symmetry group describes all symmetries of objects. ... In mathematics, the origin of a coordinate system is the point where the axes of the system intersect. ...


Real coordinate space

Let R denote the field of real numbers. For any non-negative integer n, the space of all n-tuples of real numbers forms an n-dimensional vector space over R, which is denoted Rn and sometimes called real coordinate space. An element of Rn is written In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, the real numbers may be described informally in several different ways. ... The integers are commonly denoted by the above symbol. ... In mathematics, a tuple is a finite sequence of objects, that is, a list of a limited number of objects. ...

x = (x1, x2, …, xn),

where each xi is a real number. The vector space operations on Rn are defined by

mathbf{x} + mathbf{y} = (x_1 + y_1, x_2 + y_2, ldots, x_n + y_n),
a,mathbf{x} = (a x_1, a x_2, ldots, a x_n).

The vector space Rn comes with a standard basis: In linear algebra, the standard basis for an -dimensional vector space is the basis obtained by taking the basis vectors where is the vector with a in the th coordinate and elsewhere. ...

mathbf{e}_1 = (1, 0, ldots, 0),
mathbf{e}_2 = (0, 1, ldots, 0),
vdots
mathbf{e}_n = (0, 0, ldots, 1).

An arbitrary vector in Rn can then be written in the form

mathbf{x} = sum_{i=1}^n x_i mathbf{e}_i.

Rn is the prototypical example of a real n-dimensional vector space. In fact, every real n-dimensional vector space V is isomorphic to Rn. This isomorphism is not canonical, however. A choice of isomorphism is equivalent to a choice of basis for V (by looking at the image of the standard basis for Rn in V). The reason for working with arbitrary vector spaces instead of Rn is that it is often preferable to work in a coordinate-free manner (that is, without choosing a preferred basis). In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ... Canonical is an adjective derived from canon. ... In linear algebra, a basis is a minimum set of vectors that, when combined, can address every vector in a given space. ...


Euclidean structure

Euclidean space is more than just real coordinate space. In order to do Euclidean geometry one needs to be able to talk about the distances between points and the angles between lines or vectors. The natural way in which to do this is to introduce the standard inner product (also known as the dot product) on Rn, by

mathbf{x}cdotmathbf{y} = sum_{i=1}^n x_iy_i = x_1y_1+x_2y_2+cdots+x_ny_n.

The inner product of any two vectors x and y gives a real number. Furthermore, the inner product of x with itself is always nonnegative. This product allows us to define the "length" of a vector x as

|mathbf{x}| = sqrt{mathbf{x}cdotmathbf{x}} = sqrt{sum_{i=1}^{n}(x_i)^2}.

This length function satisfies the required properties of a norm and is called the Euclidean norm on Rn. The (interior) angle θ between x and y is then given by In linear algebra, functional analysis and related areas of mathematics, a norm is a function which assigns a positive length or size to all vectors in a vector space, other than the zero vector. ...

theta = cos^{-1}left(frac{mathbf{x}cdotmathbf{y}}{|mathbf{x}||mathbf{y}|}right)

where cos−1 is the arccosine function. In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...


Finally, one can use the norm to define a distance function (or metric) on Rn by For distance between people, see proxemics. ... In mathematics a metric or distance is a function which assigns a distance to elements of a set. ...

d(mathbf{x}, mathbf{y}) = |mathbf{x} - mathbf{y}| = sqrt{sum_{i=1}^n (x_i - y_i)^2}.

This distance function is called the Euclidean metric. It can be viewed as a form of the Pythagorean theorem. The Euclidean distance of two points x = (x1,...,xn) and y = (y1,...,yn) in Euclidean n-space is computed as It is the ordinary distance between the two points that one would measure with a ruler, which can be proven by repeated application of the Pythagorean theorem. ... In mathematics, the Pythagorean theorem or Pythagoras theorem is a relation in Euclidean geometry among the three sides of a right triangle. ...


Real coordinate space together with this Euclidean structure is called Euclidean space and often denoted En. (Many authors refer to Rn itself as Euclidean space, with the Euclidean structure being understood). The Euclidean structure makes En an inner product space (in fact a Hilbert space), a normed vector space, and a metric space. In mathematics, an inner product space is a vector space with additional structure, an inner product (also called a scalar product), which allows us to introduce geometrical notions such as angles and lengths of vectors. ... In mathematics, a Hilbert space is a real or complex vector space with a positive definite sesquilinear form, that is complete under its norm. ... In mathematics, with 2- or 3-dimensional vectors with real-valued entries, the idea of the length of a vector is intuitive and can easily be extended to any real vector space Rn. ... In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ...


Rotations of Euclidean space are then defined as orientation-preserving linear transformations T that preserve angles and lengths: In mathematics, an orientation on a real vector space is a choice of which ordered bases are positively oriented (or right-handed) and which are negatively oriented (or left-handed). ... In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ...

Tmathbf{x} cdot Tmathbf{y} = mathbf{x} cdot mathbf{y},
|Tmathbf{x}| = |mathbf{x}|.

In the language of matrices, rotations are special orthogonal matrices. In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ... In matrix theory, a real orthogonal matrix is a square matrix Q whose transpose is its inverse: // An orthogonal matrix is the real specialization of a unitary matrix, and thus always a normal matrix. ...


Topology of Euclidean space

Since Euclidean space is a metric space it is also a topological space with the natural topology induced by the metric. The metric topology on En is called the Euclidean topology. A set is open in the Euclidean topology if and only if it contains an open ball around each of its points. The Euclidean topology turns out to be equivalent to the product topology on Rn considered as a product of n copies of the real line R (with its standard topology). In mathematics, a metric space is a set where a notion of distance between elements of the set is defined. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... It has been suggested that this article or section be merged with Logical biconditional. ... A synonym for ball (in geometry or topology, and in any dimension) is disk (or disc Geometry In metric geometry, a ball is a set containing all points within a specified distance of a given point. ... In topology, the cartesian product of topological spaces is turned into a topological space in the following way. ... In mathematics, the real line is simply the set of real numbers. ...


An important result on the topology of Rn, that is far from superficial, is Brouwer's invariance of domain. Any subset of Rn (with its subspace topology) that is homeomorphic to another open subset of Rn is itself open. An immediate consequence of this is that Rm is not homeomorphic to Rn if mn — an intuitively "obvious" result which is nonetheless difficult to prove. Luitzen Egbertus Jan Brouwer (February 27, 1881 - December 2, 1966), usually cited as L. E. J. Brouwer, was a Dutch mathematician, a graduate of the University of Amsterdam, who worked in topology, set theory, and measure theory and complex analysis. ... Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space Rn. ... In topology and related areas of mathematics, a subspace of a topological space X is a subset S of X which is equipped with a natural topology induced from that of X called the subspace topology (or the relative topology, or the induced topology). ... This word should not be confused with homomorphism. ...


Relation to manifolds

In modern mathematics, Euclidean spaces form the prototypes for other, more complicated geometric objects. In particular, a manifold is a Hausdorff topological space that is locally homeomorphic to Euclidean space. On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ... In topology and related branches of mathematics, a Hausdorff space is a topological space in which points can be separated by neighbourhoods. ...


Euclidean n-space is the most elementary example of an n-dimensional manifold. In fact, it is a smooth manifold. For n ≠ 4, any differentiable n-manifold that is homeomorphic to Rn is also diffeomorphic to it. The surprising fact that this is not true for n = 4 was proved by Simon Donaldson in 1982; the counterexamples are called exotic (or fake) 4-spaces. In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... This word should not be confused with homomorphism. ... In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ... Simon Kirwan Donaldson, born in Cambridge in 1957, is a mathematician famous for his work on exotic four-dimensional spaces in differential geometry using instantons, and the discovery of new differential invariants. ... 1982 (MCMLXXXII) was a common year starting on Friday of the Gregorian calendar. ... In mathematics, an exotic or fake R4 is a differentiable manifold that is homeomorphic to the Euclidean space R4, but not diffeomorphic. ...


Furthermore, Euclidean space is a linear manifold. An m-dimensional linear submanifold of Rn is a Euclidean space of m dimensions embedded in it (as an affine subspace). For example, any straight line in some higher-dimensional Euclidean space is a 1-dimensional linear submanifold of that space. An affine transformation or affine map (from the Latin, affinis, connected with) between two vector spaces consists of a linear transformation followed by a translation. ...


In general, the notion of manifolds embraces both Euclidean and non-Euclidean geometry. From this point of view, the essential property of Euclidean space is that it is flat — that is, not curved. Modern physics, specifically the theory of relativity, demonstrates that our universe is not truly Euclidean. Although this is significant in theory and even in some practical problems, such as global positioning and airplane navigation, a Euclidean model can still be used to solve many other practical problems with sufficient precision. Behavior of lines with a common perpendicular in each of the three types of geometry The term non-Euclidean geometry (also spelled: non-Euclidian geometry) describes hyperbolic, elliptic and absolute geometry, which are contrasted with Euclidean geometry. ... Curvature refers to a number of loosely related concepts in different areas of geometry. ... The first few hydrogen atom electron orbitals shown as cross-sections with color-coded probability density. ... Two-dimensional analogy of space-time distortion described in General Relativity. ... GPS satellite in orbit, image courtesy NASA GPS redirects here. ... Fixed-wing aircraft is a term used to refer to what are more commonly known as aeroplanes in Commonwealth English (excluding Canada) or airplanes in North American English. ... Table of geography, hydrography, and navigation, from the 1728 Cyclopaedia. ...


See also

In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ... In differential geometry, Riemannian geometry is the study of smooth manifolds with Riemannian metrics, i. ...

References

  • Kelley, John L. (1975). General Topology. Springer-Verlag. ISBN 0-387-90125-6.
  • Munkres, James (1999). Topology. Prentice-Hall. ISBN 0-13-181629-2.

  Results from FactBites:
 
PlanetMath: Euclidean space (148 words)
The difference between Euclidean space and a Euclidean vector space is one of loss of structure.
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